A high-accuracy extraction of the isoscalar πn scattering length from pionic deuterium data

Transcription

1 A high-accuracy extraction of the isoscalar πn scattering length from pionic deuterium data Daniel Phillips Ohio University and Universität Bonn with V. Baru, C. Hanhart, M. Hoferichter, B. Kubis, A. Nogga (JOB collaboration) Research supported by the US Department of Energy and the Deutsches Forschungsgemeinschaft

2 Tony and πd/πn scattering CLOUDY

3 πn scattering lengths In isospin limit: t ab πn(0) = 4π(1 + M π /M N )(a + δ ab + a iɛ abc τ c ) a - and a +, isoscalar and isovector scattering lengths a = M π 8πF 2 π ; a + =0 Weinberg (1969) Leading-order in an expansion in Mπ, an expansion around the chiral limit of QCD a + encodes extent of chiral-symmetry breaking Connected to other issues in strong interactions, πn sigma term, πnn coupling constant (via GMO sum rule)

4 Hadronic atoms E ns π A = α2 emµ πa 2n 2 ψ n (0) 2 2πa πa µ πa Atom is bound by Coulomb force, so characteristic distances are Mπ αem. Forms basis for expansion in powers of αem. In this theory πa scattering length is high-energy information. E 1s = E QED 1s 2α 3 emµ 2 πaa π A(1 + 2α(1 log α)µ πa a π A + δ vac A ), e.g. Lyubovitsky, Rusetsky (2000) Difference E1s-E1s QED can now be measured to 1% or better

5 Hadronic-atom observables Here focus will be on π - p and π - d atoms Can measure strong level shift, E1s-E1s QED, in both Information on scattering length for π - p, real part of scattering length for π - d Re aπd 2a + + athree-body, Can measure width in both atoms too Proton gives π - p π 0 n, pionic deuterium width gives little information on threshold πn scattering. Three constraints on a - and a + Test consistency, improve accuracy of extraction

6 How accurate? Fits to low-energy πn data give: C.f. and a + =0± M 1 π a LO = M 1 π Re a πd M 1 π Fettes, Meissner (2000) For a + to accuracy 1 x 10-3 need theory that addresses few-body dynamics on 5% level or better At this level of accuracy we need to also worry about isospin violation: from both mu-md and electromagnetic effects Claim: the calculation of aπd I will present is accurate at the 5% level, i.e. a πd M 1 π

7 Plan Introduction A theory of the πd scattering length Including isospin violation Results

8 Wanted: theory of aπd One approach: multiple-scattering series. Expansion in Here, χpt: nominally an expansion in Mπ/(4 π Fπ) Mπ/MN We incorporate isospin violation in both the twobody (πn) and three-body (πnn) sector. Counting e Mπ/MN P 4πa r d 0.35 Challenge: additional momentum scales E.g. 1/rd (MN Bd) 1/2, (Mπ Bd) 1/2

9 Tony s favourite quote(s): A foolish consistency is the hobgoblin of small minds Ralph Waldo Emerson Self-reliance Goya, from Wikipedia Commons modified by Phillips The nucleon has a finite size, and if you d just use the cloudy-bag model...

10 The simple bit... Up to O(P 3 ) Weinberg (1992), Beane et al. (1998) a πd =2 1+M π/m N 1+M π /M d a + + a (d1 ) + a (d4 ) + a (d5 ) a (d1 ) = Mπ π 4 Fπ(1 4 + M π /M d ) q 2 a(d1) 20 x 10-3 Mπ -1 =almost all of experimental aπd. Means that we must look for a + in what s left.

11 Going further Beane et al. (2003) Diagrams of O(P 4 ) contributing to threebody part of aπd : Sum to zero!

12 Tony s favourite questions: But you haven t accounted for the NN intermediate state, and it s known that the dispersive contribution is an important effect in this reaction. Use successful χpt calculation of π - d nn to calculate dispersive contribution. (Explains Im aπd.) You have a huge resonance in πn scattering only a few hundred MeV above threshold, which your calculation completely ignores. T Include effects of Delta(1232) intermediate states a disp+ =( 0.6 ± 1.5) 10 3 M 1 π. Nominally these pieces are of order P 9/2 Baru et al. (2006, 2007)

13 More on πnn mechanisms Triple scattering: a (d3 ) = (2.6 ± 0.5) 10 3 M 1 π Nominally of O(P 5 ), but enhanced by a factor of π 2 Therefore must be included to achieve desired accuracy Also need to include recoil corrections to double-scattering graph, and impact of embedding πn amplitude in πnn system. a str =( 22.7 ± 1.1 ± 0.4) 10 3 M 1 π. CD-Bonn, AV18, NNLO χet Difficult to go further: contact term at O(P 5 ) 1 x 10-3 Mπ -1

14 Isospin violation in πnn parts Pion-mass differences already incorporated in this number As were IV differences in the πn scattering lengths (only give 1% of a(d1)) Isospin violation in O(P 4 ) below our accuracy threshold But still need to compute photon exchange in three-body system, e.g. Dominant effect from Coulomb photons Need to worry about NN interaction in intermediate state

15 Scales, scales everywhere... d6, d9, and d10 potentially infrared divergent, <1/ q 4 > But regulated at hadronic atom scale: give part of level shift Need to compute part not already included in modified Deser formula (Mπ Bd) 1/2 appears thanks to three-body dynamics (MN Bd) 1/2 from deuteron wave function So, in the infra-red enhanced diagrams, does χpt counting apply?

16 A very useful theorem Baru, Epelbaum, Rusetsky (2009) Consider isoscalar part of, e.g.: Add NN interactions in int. state: ψ d T πn G NN ( B at ) ( Q k 2 ) ψ d = n ψ d T (s) πn ψ n ψ n Q k ψ 2 d T (s) πn 1 B at E n B d B at Since TπN and Q are spin and isospin independent, if no momentum transfer, orthogonality gives zero So no contribution from momenta of order (Mπ Bd) 1/2 Momenta of order (MN Bd) 1/2 give shift in aπd of < 1% a + Momenta of order Mπ give smaller contribution still from this isoscalar part k 2

17 P waves Isovector part ends up giving main contribution Only effect from momenta of order Mπ a EM = (0.95 ± 0.01) 10 3 M 1 π Consistent with power-counting estimate This is largest IV πnn mechanism So no need to consider more complex photon exchanges

18 Now the πn isospin violation... In fact, a + cannot be measured directly in either πd or πn scattering. Can only be measured in combination: { ã + a + 1 4(M 2 + π M 2 } π 0 ) c 1 2e 2 f 1 4π(1 + M π /M p ) a + is defined in the isospin limit: this corrects for mu-md and presence of hard photons F 2 π In aπd what is measured is actually, up to O(P 3 ) Hoferichter, Kubis, Meissner (2009) a (2) π d = 2(1 + M π/m p ) (ã+ + ã +), 1+M π /M d ã + = 1 4π(1 + M π /M p ) [ g 2 A M π 32πF 2 π ( 33(M 2 π M 2 π 0 ) 4F 2 π ) + e 2 + e 2( 2g6 r + g8 r ) ] Estimate: ã + =( 3.3 ± 0.3) 10 3 M 1 π

19 Numbers Experiment: Gotta et al. (2005, 2010) ɛ H 1s = ( ± 0.012) ev, Γ 1s = (0.823 ± 0.019) ev, ɛ D 1s = (2.325 ± 0.031) ev ɛ D 1s a, a cex π p a disp+ ã + Wave-function averages 16 % 21 % 75 % 30 % 53 %

20 Conclusions ã + = (2.1 ± 1.0) 10 3 Mπ 1, a = (86.3 ± 1.0) 10 3 Mπ 1 With estimates for LECs we get: a + = Mπ 1 Note theory error still larger than experiment, but difficult to do better without better knowledge of (other) shortdistance physics in both πnn and πn sectors Need to treat three-body dynamics and isospin violation carefully to achieve this level of accuracy Systematicity of χpt useful Lots of modifications to power-counting estimates

21 The counting is more like what you d call guidelines than actual rules Pirates of the Cloudy Bag: THe Curse of the Blocked Pion Pirates of the Cloudy Bag: On Stranger Quarks Pirates of the Cloudy Bag: At World s End Happy Birthday Tony!